Estimation of the mass density of biological matter from refractive index measurements

The quantification of physical properties of biological matter gives rise to novel ways of understanding functional mechanisms. One of the basic biophysical properties is the mass density (MD). It affects the dynamics in sub-cellular compartments and plays a major role in defining the opto-acoustical properties of cells and tissues. As such, the MD can be connected to the refractive index (RI) via the well known Lorentz-Lorenz relation, which takes into account the polarizability of matter. However, computing the MD based on RI measurements poses a challenge, as it requires detailed knowledge of the biochemical composition of the sample. Here we propose a methodology on how to account for assumptions about the biochemical composition of the sample and respective RI measurements. To this aim, we employ the Biot mixing rule of RIs alongside the assumption of volume additivity to find an approximate relation of MD and RI. We use Monte-Carlo simulations and Gaussian propagation of uncertainty to obtain approximate analytical solutions for the respective uncertainties of MD and RI. We validate this approach by applying it to a set of well-characterized complex mixtures given by bovine milk and intralipid emulsion and employ it to estimate the MD of living zebrafish (Danio rerio) larvae trunk tissue. Our results illustrate the importance of implementing this methodology not only for MD estimations but for many other related biophysical problems, such as mechanical measurements using Brillouin microscopy and transient optical coherence elastography.

Supporting Material: Estimations of the mass density of biological matter from refractive index measurements

I. PARTIAL AND APPARENT SPECIFIC VOLUMES AND THE REFRACTIVE INDEX INCREMENT
Based on derivations in [1], for a binary solution consistent of N 1 moles of solvent with molar volume V 1 and N 2 moles of solute, the total molar volume is given by where Θ denotes the apparent molar volume of the solute.This parameter accounts for the potential volume-inadditivity; in other words, it is a macroscopic quantity that describes the volume of the solute in dilution.The partial molar volume is given by As employed in [1][2][3][4], the PSV and refraction per gram of a protein may be interpreted the weight average of all amino acids composing this protein as where M w = 18.01 g/mol is the molecular mass of water [5,6] and N A is the Avogadro constant.We note that, without loss of generality, setting m i = M i /N A , i.e. n i = 1/N A , Eq. (S4) can be expressed as which coincides with the Lorentz-Lorenz mixing rule.This can be seen by where a i is the respective mole fraction.• Errors in the PSV considerations -The following effects are stated in [1] * conformational protein formation from amino acids: ∼ 0.012 ml/g * volume changes of proteins in solution: • swelling and dissolution: ∼ −5 × 10 −2 ml/g • thermal denaturation: ∼ ±10 −2 ml/g • Coil-helix transition: ∼ +10 −2 ml/g • Aggregation: ∼ +5 × 10 −3 ml/g • Sol-gel transition: ∼ −10 −4 ml/g • Errors in the RI increment considerations - [4] found that the predicted RI increment values using the Wiener relation for different proteins are systematically higher than the experimental values.Further they found a linear relationship between predicted and experimental values.Further they propose a π − π interaction term correction which slightly improves the agreement.We do not employ this correction here.However, we note that this discrepancy could also be partially resolved by not employing the Wiener equation but rather relying on the Biot equation, as well as a different choice of experimental values of R * r for the amino acid residues.

D. Employed amino acid residues values
We omitted the contributions of selenocysteine since we could not obtain data on molar refractivity, mass density nor refractive index.For all other amino acid residues (AAR) the employed values are given in table S1.

II. PARAMETER VALUES
In the following we provide the mean values ± standard deviations of the following quantities; relative volume fraction x, refractive index n, refraction per gram R in cm 3 /g, PSV θ in ml/g and RI increment α in ml/g of the different macro molecules employed throughout this study in Table S2.The RI and refraction per gram values of the lipids employed in the sections Lipids and proteins in water and Larval zebrafish trunk tissue were obtained from [8] (predicted data by ACD/Labs Percepta Platform -PhysChem Module, at T = 20 °C and a wavelength of λ = 589 nm), from which we computed the PSV via the Lorentz-Lorenz relation (Eq.( 10)).The associated standard deviation was then estimated via Gaussian propagation of uncertainty.For all the computations presented in this study, we interpret a value that has a uncertainty attached to it as a normal distribution, where the standard deviation is given by said uncertainty.For values that were assumed to be precise, we assume a delta distribtuion.TABLE S2: Rounded mean values ± standard deviations of relative volume fraction x, refractive index n, refraction per gram R in cm 3 /g, PSV θ in ml/g and RI increment α in ml/g of the different macro molecules employed throughout this study.Entries without footnote indicate assumptions or derived values (see main text for further information).Assumed to be precise values are stated without standard deviation.b obtained from [8] c obtained from [10] g obtained from the manufacturer h obtained from [11] i obtained from [12] j obtained from [13] d computed from values obtained from [14] e computed from AA sequences obtained from [15] and [9] f computed from values obtained from [16] III.REFRACTIVE INDEX MIXING MODEL COMPARISONS Following [17,18], assuming volume additivity, we may express the different mixing rules as where the different f i are specific for each mixing rule, ϕ s and n s denotes the solute volume fraction and RI, respectively.The f i under study are listed in table S3.By solving Eqs.(S7) of the respective mixing rule for the mixture RI, we obtain a functional relationship n(n s , c s , ρ s ) that can be used to fit experimental data, as shown in Fig. 4A.Since we obtained the solute MD ρ s from measurements of the mixture MD in dependence of c s (see Fig. 4B), by fitting the data with n(n s , c s , ρ s ), we obtain a value of solute RI n s for each model.We then evaluate the difference between the respective fitted solute RIs and the -based on the biochemical composition-predicted solute RI of the samples under study, using different mixing rules.The results are given in Table S3.TABLE S3: f i as defined in Eq. (S7) for different mixing rules and difference between fitted and predicted solute RIs n fit s − n pred s for the respective RI mixing rules.The uncertainty intervals were computed employing Gaussian propagation of uncertainty.The predicted RI distributions were obtain as outlined in the main text for N 0 = 10 3 and N v = 10 3 .RI Mixture rules were adapted from [17].Following [18], assuming volume additivity, the Lorentz-Lorenz mixing rule of RIs can be expressed as Next, we solve above equation for n LL to obtain from which we compute the RI increment α LL ≡ ∂n LL /∂c 2 as Now, Eq.(S10) has to be solved for the solute concentration c 2 .While this is analytically possible, the resulting expression is quite lengthy, so we refer to the resulting expression as c 2 (α LL , n 1 , n 2 , θ).Of course, this solving step can be done numerically, given a set of parameter values.Finally, following the rational of the main text, we insert the resulting expression for the solute concentration in Eq. ( 6) to obtain We note that all other considerations presented in the main text can be adapted for Eq.(S11), while the results will be, unfortunately, more lengthy and difficult to handle.

V. PDF OF A TRUNCATED NORMAL DISTRIBUTION T
For a normal distribution with mean µ and standard deviation σ, the corresponding truncated distribution is denoted by T (µ, σ).The probability density function (PDF) of T is defined as where f (y) and F (y) denote the PDF and cumulative distribution function (CDF), respectively, of said normal distribution N (µ, σ).

VI. MEAN AND STANDARD DEVIATION OF NORMAL MIXTURE DISTRIBUTIONS
Let's consider a one dimensional mixture distribution of two Normal distributions, with 0 ≤ w ≤ 1.The mean and standard deviation of this mixture distribtion is given by respectively.For the special case of σ 1 = σ 2 = σ we obtain for the standard deviation of the mixture distribution for all µ 1 = µ 2 and 0 ≤ w ≤ 1, which is maximized for w = 0.5.

VII. EFFECTIVE PARAMETERS
From the mixing rule of the solute density we obtain that the effective PSV is where N s is the nuber of solute voxelinos per voxel and j denotes the different types of solute molecules.The effective RI increment for the Biot equation is then (S18)

VIII. UNCERTAINTIES OF THE EFFECTIVE PARAMETERS FOR LIPIDS AND PROTEINS IN WATER
The uncertainties of the effective RI increment and the PSV, introduced in Eq. ( 21) can be computed as follows.The terms ∆α 0 eff and ∆θ 0 eff refer to the standard deviations of the mixture distribution, as given in Eq. (S14).The uncertainties associated of the effective PSV to deviations in the relative lipid volume fraction ∆x lip , can be evaluated by employing Eq. ( 16), from which we obtain that We may write the effective RI increment, using Eq. ( 16), as Hence, we have IX. REFRACTIVE INDEX VALUES OF LARVAL ZEBRAFISH TRUNK TISSUE AT 96 PF FROM [15] The RI values of larval zebrafish trunk tissue at 96 hpf employed in this study are not explicitly stated in [15], hence we list them here.Based on the measurements of [14], where they determined the wet mass m tot , dry mass m dry , protein mass m p and the lipid mass m lip of larval zebrafish at 96 hpf, we define the relative lipid mass fraction y lip ≡ m lip /(m lip + m p ) and consider the ratio from which we obtain and analogously Based on this dry mass composition we estimate the distributions of θ eff p and θ eff lip by running the MC simulation of the extended mixture model with x lip = 0 and x lip = 1, respectively.We may now obtain the distribution of x lip by a MC sampling approach, considering the distributions of all individual parameters.Next, we compute θ eff dry by repeating the procedure described above for x lip following the distribution determined previously.With that we obtain the distribution of ϕ 1 .

XI. RELATIVE UNCERTAINTY OF THE RI FOR DIFFERENT MATERIAL PROPERTIES
The dependence of the relative RI deviation on the deviation of the relative lipid volume fraction ∆x lip for different deviations of the water volume fraction ∆ϕ 1 , as pointed out in the main text, can be qualitatively understood with

XII. STRATEGIES FOR ESTIMATING THE MD FOR CERTAIN EXPERIMENTAL PARADIGMS
In Fig. S2, we outline possible strategies for estimating the MD, given certain experimental insights.In the following, we use the same abbreviations as in the main text, namely, • RI = refractive index, • (S)RS = (stimulated) Raman spectroscopy, • MS = mass spectrometetry.
We want to point out that the estimation process is heavily dependent on identifying relevant solute constituents of the sample, as well as their PSVs, RIs and/or refractions per gram.For the latter, secondary data bases, such as ChemSpider [8], are invaluable.Furthermore, although RI measurements are not necessary to predict ρ(δn), as outlined in the main text, comparing predicted and measured RI distributions gives necessary insight on the validity of the prediction.Strategy: 1) Try to obtain as much information as possible about the biochemical composition of the sample under study.See e.g., the main text for the case of trunk tissue of the larval zebra fish.
2) Try to obtain or compute the θj, Rj, or nj of all the types of solute constituents (e.g., proteins, lipids, sugars, etc.) identified in step 1).
3) Sample Eqs. ( 7) and ( 14 FIG. S1: Asymptotic relative deviation of the RI (∆n) ∞ /n for different potential differences of lipid and protein PSVs θ lip − θ p in dependence of the difference of lipid and protein refractions per gram R lip − R p for the cases of a non-fluctuating water volume fraction ∆ϕ 1 = 0 (top) and a fluctuating water volume fraction ∆ϕ 1 = 0.1 (bottom).
FIG. S2: Flowchart of mass density estimation approaches outlined in this study, based on different experimental paradigms.The abbreviations and symbols are as defined in the main text.
C. Errors with assuming volume additivity g [9]omputed from AA sequences obtained from[9]

TABLE S4 :
[15]active indices n and according standard deviations ∆n of the trunk tissue of N = 20 zebrafish larvae at 96 hpf from[15].
X. ESTIMATIONS OF x lip AND ϕ1 OF LARVAL ZEBRAFISH AT 96 HPF